Further Bijections to Pattern-Avoiding Valid Hook Configurations

TL;DR

This paper explores bijections involving valid hook configurations related to pattern-avoiding permutations, connecting them to Motzkin paths and lattice walks, and resolving enumeration questions about their generating functions and asymptotics.

Contribution

It extends bijections to intervals in Motzkin posets and proves that valid hook configurations for 312-avoiding permutations are not D-finite, also providing a new proof for 132-avoiding cases.

Findings

Valid hook configurations for 312-avoiding permutations are not D-finite.

Established bijections between valid hook configurations and lattice paths.

Provided asymptotic enumeration results for these configurations.

Abstract

Valid hook configurations are combinatorial objects used to understand West's stack-sorting map. We extend existing bijections corresponding valid hook configurations to intervals in partial orders on Motzkin paths. To enumerate valid hook configurations on $312$-avoiding permutations, we build off of an existing bijection into a Motzkin poset and construct a bijection to certain well-studied closed lattice walks in the first quadrant. We use existing results about these lattice paths to show that valid hook configurations on $312$-avoiding permutations are not counted by a $D$-finite generating function, resolving a question of Defant's, and additionally to compute asymptotics for the number of such configurations. We also extend a bijection of Defant's to a correspondence between valid hook configurations on $132$-avoiding permutations and intervals in the Motzkin-Tamari posets, providing a more elegant proof of Defant's enumeration thereof. To investigate this bijection, we present a number of lemmas about valid hook configurations that are generally applicable and further study the bijections of Defant's.